Discussion on the Central Limit Theorem: Conditions and Applications

This discussion post delves into the Central Limit Theorem (CLT), outlining the conditions necessary for its application, including random data distribution, independent samples, a sample size less than 10% of the population, and a sufficiently large sample size. It identifies the mean as the key statistical quantity applicable to the CLT and explains how the theorem can be used in future career for data analysis, particularly in scenarios with large populations where sampling and averaging can approximate a normal distribution. The post references external sources to support its explanations and provides a practical example of using the CLT in election opinion polls to predict outcomes based on sample data.

Conditions for Using the Central Limit Theorem
By definition, the central limit theorem posits that, given some conditions, “…as the size of the
sample increases, the distribution of the mean across multiple samples will approximate a
Gaussian distribution.” (Brownlee, 2018). According to a post on
https://www.youtube.com/watch?v=jvoxEYmQHNM, the central limit theorem follows the
conditions that:
i. Data must be randomly distributed
ii. Samples should be independent of each other
iii. The sample size should be less than 10% of the population
iv. Population Sample size should be sufficiently large
Generally, given the above discussion, it can be argued that based on the central limit theorem,
the averages of samples from a given population have an approximately normal distribution.
Moreover, taking into consideration the condition of CLT that the data ought to be randomly
distributed and the sample size should be sufficiently large the following can be interpreted:
Data should be randomly distributed and sample size chosen be sufficiently large
In the central limit theorem, selecting a sufficiently large sample from the population i.e. a
sample of size 30 and above is sufficient to hold for the CLT regardless of whether the original
population is from a normal distribution or not. In the event that the original population is from a
normal distribution, the central limit theorem will hold regardless of the sample size i.e. the
sample means will be normally distributed even for a sample whose size is less than 30.
What statistical quantity is applicable for applying the Central Limit Theorem?
Mainly, among the five basic statistical quantities which include: Mean, Median, Mode,
Variance and standard deviation; in the CLT, the mean is the basic statistical quantity used in its

application. This can be inferred from the definition of CLT by Chen, Goldstein & Shao (2011)
i.e. “if you have a population with mean μ and standard deviation σ and take sufficiently large
random samples from the population with replacement”, the means of the selected samples are
bound to follow a normal distribution. (Chen, Goldstein, & Shao, 2011).
Explain how you can apply the Central Limit Theorem to analyzing data in your future
career
Ideally, according to the definition of the central limit theorem, if one selects random samples
from a given population and computes the mean of each sample, then such averages will be
approximated to a normal distribution.
In real life, quite a number of occasions require data analysis. However, in several situations the
target population involved in the process of data collection might be relatively large which tends
to increase the cost of data collection and the time required. It is in such situations that the
central limit theorem can be applied. For instance, it is a common practice that during the period
preceding elections, opinion polls are conducted by different companies. In this regard, in case
we would like guess the results of an upcoming election we might consider taking a poll. In our
results we might find that 60% of the voters would vote for A instead of B. For us to determine if
these results apply to the whole population, we can apply the central limit theorem and conduct
the sample poll again and again and guess who will win given that the entire voter population
votes.
References
Brownlee, J. (2018, May 4). A Gentle Introduction to the Central Limit Theorem for Machine
Learning. Retrieved from Machine Learning Mastery:

https://machinelearningmastery.com/a-gentle-introduction-to-the-central-limit-theorem-
for-machine-learning/
Chen, L. H., Goldstein, L., & Shao, Q. M. (2011). Normal approximation by Stein's method.
New York: Springer.